Monthly pension returns analysis: Velliv June 2012 - April 2024
Martin Hoshi Vognsen
08:53 05 June 2024
- Returns data
- Compare pension plans
- Compare Gaussian and skewed t-distribution fits
- Velliv medium risk (vmr), June 2012 - April 2024
- Velliv high risk (vhr), June 2012 - April 2024
- PFA medium risk (pmr), June 2012 - April 2024
- PFA high risk (phr), June 2012 - April 2024
- Mix medium risk (mmr), June 2012 - April 2024
- Mix high risk (mhr), June 2012 - April 2024
- Mix vmr+phr (vm_ph), June 2012 - April 2024
- Mix vhr+pmr (mh_pm), June 2012 - April 2024
- Velliv medium risk (vmr), June 2012 - April 2024
Fit log returns to F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of
freedom).
xi is the estimated skewness parameter.
Returns data
The long version of Velliv medium risk data runs from January 2007 to
April 2024 (incl).
For January 2007 to May 2012 no low risk and high risk funds existed.
For this period the medium risk data is copied into the other funds.
The short version runs from June 2012 to April 2024.
Velliv returns are including bonus and “DinKapital.
PFA returns are including”KundeKapital”.
Summary of log-returns
The summary statistics are transformed back to the scale of gross returns by taking \(exp()\) of each summary statistic. (Note: Taking arithmetic mean of gross returns directly is no good. Must be geometric mean.)
| vmr | vhr | vmrl | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|---|
| Min. : | 0.901 | 0.877 | 0.901 | 0.924 | 0.886 | 0.912 | 0.882 | 0.893 | 0.899 |
| 1st Qu.: | 0.996 | 0.994 | 0.995 | 0.999 | 0.994 | 0.997 | 0.995 | 0.996 | 0.996 |
| Median : | 1.010 | 1.012 | 1.007 | 1.008 | 1.012 | 1.009 | 1.013 | 1.011 | 1.011 |
| Mean : | 1.006 | 1.007 | 1.005 | 1.005 | 1.008 | 1.006 | 1.008 | 1.007 | 1.006 |
| 3rd Qu.: | 1.021 | 1.027 | 1.020 | 1.015 | 1.025 | 1.018 | 1.025 | 1.022 | 1.021 |
| Max. : | 1.070 | 1.088 | 1.070 | 1.043 | 1.079 | 1.054 | 1.082 | 1.073 | 1.065 |
Ranking
| Min. : | ranking | 1st Qu.: | ranking | Median : | ranking | Mean : | ranking | 3rd Qu.: | ranking | Max. : | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.924 | pmr | 0.999 | pmr | 1.013 | mhr | 1.008 | phr | 1.027 | vhr | 1.088 | vhr |
| 0.912 | mmr | 0.997 | mmr | 1.012 | phr | 1.008 | mhr | 1.025 | phr | 1.082 | mhr |
| 0.901 | vmr | 0.996 | vmr | 1.012 | vhr | 1.007 | vhr | 1.025 | mhr | 1.079 | phr |
| 0.901 | vmrl | 0.996 | vhr_pmr | 1.011 | vmr_phr | 1.007 | vmr_phr | 1.022 | vmr_phr | 1.073 | vmr_phr |
| 0.899 | vhr_pmr | 0.996 | vmr_phr | 1.011 | vhr_pmr | 1.006 | vhr_pmr | 1.021 | vmr | 1.070 | vmr |
| 0.893 | vmr_phr | 0.995 | mhr | 1.010 | vmr | 1.006 | vmr | 1.021 | vhr_pmr | 1.070 | vmrl |
| 0.886 | phr | 0.995 | vmrl | 1.009 | mmr | 1.006 | mmr | 1.020 | vmrl | 1.065 | vhr_pmr |
| 0.882 | mhr | 0.994 | phr | 1.008 | pmr | 1.005 | pmr | 1.018 | mmr | 1.054 | mmr |
| 0.877 | vhr | 0.994 | vhr | 1.007 | vmrl | 1.005 | vmrl | 1.015 | pmr | 1.043 | pmr |
Correlations and covariance
Correlations
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 1.000 | 0.997 | 0.961 | 0.964 |
| vhr | 0.997 | 1.000 | 0.951 | 0.967 |
| pmr | 0.961 | 0.951 | 1.000 | 0.977 |
| phr | 0.964 | 0.967 | 0.977 | 1.000 |
Covariances
| vmr | vhr | pmr | phr | |
|---|---|---|---|---|
| vmr | 0.001 | 0.001 | 0 | 0.001 |
| vhr | 0.001 | 0.001 | 0 | 0.001 |
| pmr | 0.000 | 0.000 | 0 | 0.000 |
| phr | 0.001 | 0.001 | 0 | 0.001 |
Compare pension plans
Risk of loss
Risk of loss at least as big as row name in percent for a single period (year).
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 32.333 | 33.000 | 30.167 | 31.667 | 31.167 | 32.167 | 31.833 | 31.667 |
| 5 | 2.500 | 4.167 | 0.667 | 3.167 | 1.500 | 3.500 | 2.667 | 2.500 |
| 10 | 0.167 | 0.500 | 0.000 | 0.167 | 0.000 | 0.333 | 0.167 | 0.167 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 28.833 | 29.500 | 22.667 | 29.333 | 25.667 | 29.333 | 29 | 27.333 |
| 5 | 1.833 | 3.167 | 0.833 | 2.500 | 1.167 | 2.667 | 2 | 1.833 |
| 10 | 0.000 | 0.333 | 0.000 | 0.000 | 0.000 | 0.167 | 0 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 40.333 | 40.667 | 37.833 | 38.500 | 39.167 | 39.500 | 39.167 | 39.667 |
| 5 | 0.500 | 2.333 | 0.000 | 1.333 | 0.000 | 1.667 | 0.833 | 0.500 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 90 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 99 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Worst ranking for loss percentiles
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.000 | vhr | 4.167 | vhr | 0.500 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 32.333 | vmr | 3.500 | mhr | 0.333 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 32.167 | mhr | 3.167 | phr | 0.167 | vmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 31.833 | vmr_phr | 2.667 | vmr_phr | 0.167 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 31.667 | phr | 2.500 | vmr | 0.167 | vmr_phr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 31.667 | vhr_pmr | 2.500 | vhr_pmr | 0.167 | vhr_pmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 31.167 | mmr | 1.500 | mmr | 0.000 | pmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 30.167 | pmr | 0.667 | pmr | 0.000 | mmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 29.500 | vhr | 3.167 | vhr | 0.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 29.333 | phr | 2.667 | mhr | 0.167 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 29.333 | mhr | 2.500 | phr | 0.000 | vmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 29.000 | vmr_phr | 2.000 | vmr_phr | 0.000 | pmr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 28.833 | vmr | 1.833 | vmr | 0.000 | phr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 27.333 | vhr_pmr | 1.833 | vhr_pmr | 0.000 | mmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 25.667 | mmr | 1.167 | mmr | 0.000 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 22.667 | pmr | 0.833 | pmr | 0.000 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 40.667 | vhr | 2.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 40.333 | vmr | 1.667 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 39.667 | vhr_pmr | 1.333 | phr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 39.500 | mhr | 0.833 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 39.167 | mmr | 0.500 | vmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 39.167 | vmr_phr | 0.500 | vhr_pmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 38.500 | phr | 0.000 | pmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 37.833 | pmr | 0.000 | mmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Chance of min gains
Chance of gains of at least x percent for a single
period (year).
x values are row names.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 67.667 | 67.000 | 69.833 | 68.333 | 68.833 | 67.833 | 68.167 | 68.333 |
| 5 | 1.167 | 3.833 | 0.167 | 3.667 | 0.500 | 3.333 | 2.167 | 1.333 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 71.167 | 70.500 | 77.333 | 70.667 | 74.333 | 70.667 | 71.000 | 72.667 |
| 5 | 7.833 | 13.500 | 3.667 | 11.500 | 5.500 | 12.500 | 9.667 | 8.167 |
| 10 | 0.667 | 1.833 | 0.167 | 1.167 | 0.333 | 1.333 | 0.833 | 0.833 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 59.667 | 59.333 | 62.167 | 61.500 | 60.833 | 60.5 | 60.833 | 60.333 |
| 5 | 3.333 | 8.333 | 0.167 | 7.167 | 1.333 | 7.5 | 5.167 | 3.500 |
| 10 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 25 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 50 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
| 100 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.0 | 0.000 | 0.000 |
Best ranking for gains percentiles
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.833 | pmr | 3.833 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 68.833 | mmr | 3.667 | phr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 68.333 | phr | 3.333 | mhr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 68.333 | vhr_pmr | 2.167 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 68.167 | vmr_phr | 1.333 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 67.833 | mhr | 1.167 | vmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 67.667 | vmr | 0.500 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 67.000 | vhr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.333 | pmr | 13.500 | vhr | 1.833 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 74.333 | mmr | 12.500 | mhr | 1.333 | mhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 72.667 | vhr_pmr | 11.500 | phr | 1.167 | phr | 0 | pmr | 0 | pmr | 0 | pmr |
| 71.167 | vmr | 9.667 | vmr_phr | 0.833 | vmr_phr | 0 | phr | 0 | phr | 0 | phr |
| 71.000 | vmr_phr | 8.167 | vhr_pmr | 0.833 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 70.667 | phr | 7.833 | vmr | 0.667 | vmr | 0 | mhr | 0 | mhr | 0 | mhr |
| 70.667 | mhr | 5.500 | mmr | 0.333 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 70.500 | vhr | 3.667 | pmr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 62.167 | pmr | 8.333 | vhr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 61.500 | phr | 7.500 | mhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 60.833 | mmr | 7.167 | phr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 60.833 | vmr_phr | 5.167 | vmr_phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 60.500 | mhr | 3.500 | vhr_pmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 60.333 | vhr_pmr | 3.333 | vmr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 59.667 | vmr | 1.333 | mmr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 59.333 | vhr | 0.167 | pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
MC risk percentiles
Risk of loss at least as big as row name in percent from first to last period.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 17.71 | 18.08 | 10.00 | 11.89 | 7.65 | 7.23 | 7.11 | 8.27 |
| 5 | 9.25 | 11.04 | 3.51 | 6.44 | 2.25 | 2.58 | 2.60 | 2.78 |
| 10 | 4.54 | 6.57 | 1.19 | 3.42 | 0.62 | 0.90 | 0.75 | 0.85 |
| 25 | 0.47 | 0.88 | 0.09 | 0.35 | 0.02 | 0.02 | 0.04 | 0.03 |
| 50 | 0.02 | 0.03 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 4.21 | 4.55 | 2.33 | 3.44 | 0.64 | 0.58 | 0.60 | 0.62 |
| 5 | 2.12 | 2.64 | 1.13 | 1.67 | 0.30 | 0.18 | 0.18 | 0.23 |
| 10 | 1.08 | 1.32 | 0.57 | 0.73 | 0.11 | 0.11 | 0.08 | 0.06 |
| 25 | 0.10 | 0.21 | 0.15 | 0.06 | 0.02 | 0.01 | 0.00 | 0.01 |
| 50 | 0.00 | 0.00 | 0.04 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 14.05 | 15.07 | 9.03 | 9.83 | 4.83 | 4.39 | 4.46 | 5.75 |
| 5 | 6.01 | 8.08 | 2.22 | 4.34 | 0.64 | 1.03 | 1.01 | 1.15 |
| 10 | 1.97 | 3.53 | 0.29 | 1.66 | 0.02 | 0.23 | 0.14 | 0.12 |
| 25 | 0.00 | 0.07 | 0.00 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |
| 50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 90 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 99 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Worst ranking for MC loss percentiles
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.08 | vhr | 11.04 | vhr | 6.57 | vhr | 0.88 | vhr | 0.03 | vhr | 0 | vmr | 0 | vmr |
| 17.71 | vmr | 9.25 | vmr | 4.54 | vmr | 0.47 | vmr | 0.02 | vmr | 0 | vhr | 0 | vhr |
| 11.89 | phr | 6.44 | phr | 3.42 | phr | 0.35 | phr | 0.01 | pmr | 0 | pmr | 0 | pmr |
| 10.00 | pmr | 3.51 | pmr | 1.19 | pmr | 0.09 | pmr | 0.00 | phr | 0 | phr | 0 | phr |
| 8.27 | vhr_pmr | 2.78 | vhr_pmr | 0.90 | mhr | 0.04 | vmr_phr | 0.00 | mmr | 0 | mmr | 0 | mmr |
| 7.65 | mmr | 2.60 | vmr_phr | 0.85 | vhr_pmr | 0.03 | vhr_pmr | 0.00 | mhr | 0 | mhr | 0 | mhr |
| 7.23 | mhr | 2.58 | mhr | 0.75 | vmr_phr | 0.02 | mmr | 0.00 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 7.11 | vmr_phr | 2.25 | mmr | 0.62 | mmr | 0.02 | mhr | 0.00 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.55 | vhr | 2.64 | vhr | 1.32 | vhr | 0.21 | vhr | 0.04 | pmr | 0.01 | pmr | 0 | vmr |
| 4.21 | vmr | 2.12 | vmr | 1.08 | vmr | 0.15 | pmr | 0.00 | vmr | 0.00 | vmr | 0 | vhr |
| 3.44 | phr | 1.67 | phr | 0.73 | phr | 0.10 | vmr | 0.00 | vhr | 0.00 | vhr | 0 | pmr |
| 2.33 | pmr | 1.13 | pmr | 0.57 | pmr | 0.06 | phr | 0.00 | phr | 0.00 | phr | 0 | phr |
| 0.64 | mmr | 0.30 | mmr | 0.11 | mmr | 0.02 | mmr | 0.00 | mmr | 0.00 | mmr | 0 | mmr |
| 0.62 | vhr_pmr | 0.23 | vhr_pmr | 0.11 | mhr | 0.01 | mhr | 0.00 | mhr | 0.00 | mhr | 0 | mhr |
| 0.60 | vmr_phr | 0.18 | mhr | 0.08 | vmr_phr | 0.01 | vhr_pmr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0 | vmr_phr |
| 0.58 | mhr | 0.18 | vmr_phr | 0.06 | vhr_pmr | 0.00 | vmr_phr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 90 | ranking | 99 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.07 | vhr | 8.08 | vhr | 3.53 | vhr | 0.07 | vhr | 0 | vmr | 0 | vmr | 0 | vmr |
| 14.05 | vmr | 6.01 | vmr | 1.97 | vmr | 0.02 | phr | 0 | vhr | 0 | vhr | 0 | vhr |
| 9.83 | phr | 4.34 | phr | 1.66 | phr | 0.00 | vmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 9.03 | pmr | 2.22 | pmr | 0.29 | pmr | 0.00 | pmr | 0 | phr | 0 | phr | 0 | phr |
| 5.75 | vhr_pmr | 1.15 | vhr_pmr | 0.23 | mhr | 0.00 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 4.83 | mmr | 1.03 | mhr | 0.14 | vmr_phr | 0.00 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 4.46 | vmr_phr | 1.01 | vmr_phr | 0.12 | vhr_pmr | 0.00 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 4.39 | mhr | 0.64 | mmr | 0.02 | mmr | 0.00 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
MC gains percentiles
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 82.29 | 81.92 | 90.00 | 88.11 | 92.35 | 92.77 | 92.89 | 91.73 |
| 5 | 70.65 | 72.69 | 75.97 | 79.76 | 78.86 | 84.34 | 82.99 | 80.04 |
| 10 | 55.47 | 61.27 | 55.27 | 68.75 | 55.72 | 70.44 | 66.97 | 61.66 |
| 25 | 13.20 | 24.67 | 4.82 | 29.65 | 3.28 | 20.10 | 13.41 | 9.13 |
| 50 | 0.14 | 1.52 | 0.02 | 1.64 | 0.00 | 0.21 | 0.02 | 0.05 |
| 100 | 0.00 | 0.04 | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| 200 | 0.00 | 0.01 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 |
| 300 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 95.79 | 95.45 | 97.67 | 96.56 | 99.36 | 99.42 | 99.40 | 99.38 |
| 5 | 92.09 | 92.35 | 94.81 | 93.66 | 98.00 | 98.15 | 98.21 | 98.12 |
| 10 | 86.07 | 87.80 | 89.06 | 88.99 | 94.76 | 95.63 | 94.87 | 95.25 |
| 25 | 54.92 | 65.59 | 49.09 | 64.12 | 54.65 | 72.05 | 64.24 | 65.03 |
| 50 | 10.26 | 24.55 | 4.50 | 18.98 | 2.96 | 14.88 | 7.98 | 8.18 |
| 100 | 0.28 | 1.31 | 0.17 | 0.50 | 0.05 | 0.12 | 0.04 | 0.11 |
| 200 | 0.03 | 0.04 | 0.02 | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 |
| 300 | 0.02 | 0.02 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| 0 | 85.95 | 84.93 | 90.97 | 90.17 | 95.17 | 95.61 | 95.54 | 94.25 |
| 5 | 72.53 | 74.85 | 75.33 | 81.58 | 82.78 | 87.77 | 85.46 | 82.92 |
| 10 | 56.07 | 62.19 | 52.17 | 69.19 | 58.26 | 74.22 | 69.32 | 62.77 |
| 25 | 13.98 | 25.16 | 4.79 | 29.60 | 4.08 | 21.63 | 13.74 | 9.72 |
| 50 | 0.40 | 2.19 | 0.00 | 2.08 | 0.00 | 0.21 | 0.05 | 0.01 |
| 100 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 200 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 300 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 400 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 500 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 1000 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Best ranking for MC gains percentiles
Skewed \(t\)-distribution (sstd):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 92.89 | vmr_phr | 84.34 | mhr | 70.44 | mhr | 29.65 | phr | 1.64 | phr | 0.04 | vhr |
| 92.77 | mhr | 82.99 | vmr_phr | 68.75 | phr | 24.67 | vhr | 1.52 | vhr | 0.01 | pmr |
| 92.35 | mmr | 80.04 | vhr_pmr | 66.97 | vmr_phr | 20.10 | mhr | 0.21 | mhr | 0.01 | phr |
| 91.73 | vhr_pmr | 79.76 | phr | 61.66 | vhr_pmr | 13.41 | vmr_phr | 0.14 | vmr | 0.00 | vmr |
| 90.00 | pmr | 78.86 | mmr | 61.27 | vhr | 13.20 | vmr | 0.05 | vhr_pmr | 0.00 | mmr |
| 88.11 | phr | 75.97 | pmr | 55.72 | mmr | 9.13 | vhr_pmr | 0.02 | pmr | 0.00 | mhr |
| 82.29 | vmr | 72.69 | vhr | 55.47 | vmr | 4.82 | pmr | 0.02 | vmr_phr | 0.00 | vmr_phr |
| 81.92 | vhr | 70.65 | vmr | 55.27 | pmr | 3.28 | mmr | 0.00 | mmr | 0.00 | vhr_pmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0.01 | vhr | 0.01 | vhr | 0.01 | vhr | 0.01 | vhr | 0.01 | vhr |
| 0.01 | phr | 0.00 | vmr | 0.00 | vmr | 0.00 | vmr | 0.00 | vmr |
| 0.00 | vmr | 0.00 | pmr | 0.00 | pmr | 0.00 | pmr | 0.00 | pmr |
| 0.00 | pmr | 0.00 | phr | 0.00 | phr | 0.00 | phr | 0.00 | phr |
| 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr |
| 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr |
| 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr |
| 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr |
Standardized \(t\)-distribution (std):
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.42 | mhr | 98.21 | vmr_phr | 95.63 | mhr | 72.05 | mhr | 24.55 | vhr | 1.31 | vhr |
| 99.40 | vmr_phr | 98.15 | mhr | 95.25 | vhr_pmr | 65.59 | vhr | 18.98 | phr | 0.50 | phr |
| 99.38 | vhr_pmr | 98.12 | vhr_pmr | 94.87 | vmr_phr | 65.03 | vhr_pmr | 14.88 | mhr | 0.28 | vmr |
| 99.36 | mmr | 98.00 | mmr | 94.76 | mmr | 64.24 | vmr_phr | 10.26 | vmr | 0.17 | pmr |
| 97.67 | pmr | 94.81 | pmr | 89.06 | pmr | 64.12 | phr | 8.18 | vhr_pmr | 0.12 | mhr |
| 96.56 | phr | 93.66 | phr | 88.99 | phr | 54.92 | vmr | 7.98 | vmr_phr | 0.11 | vhr_pmr |
| 95.79 | vmr | 92.35 | vhr | 87.80 | vhr | 54.65 | mmr | 4.50 | pmr | 0.05 | mmr |
| 95.45 | vhr | 92.09 | vmr | 86.07 | vmr | 49.09 | pmr | 2.96 | mmr | 0.04 | vmr_phr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0.04 | vhr | 0.02 | vmr | 0.01 | pmr | 0.01 | pmr | 0 | vmr |
| 0.03 | vmr | 0.02 | vhr | 0.00 | vmr | 0.00 | vmr | 0 | vhr |
| 0.02 | pmr | 0.02 | pmr | 0.00 | vhr | 0.00 | vhr | 0 | pmr |
| 0.01 | mmr | 0.00 | phr | 0.00 | phr | 0.00 | phr | 0 | phr |
| 0.01 | vhr_pmr | 0.00 | mmr | 0.00 | mmr | 0.00 | mmr | 0 | mmr |
| 0.00 | phr | 0.00 | mhr | 0.00 | mhr | 0.00 | mhr | 0 | mhr |
| 0.00 | mhr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0.00 | vmr_phr | 0 | vmr_phr |
| 0.00 | vmr_phr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0.00 | vhr_pmr | 0 | vhr_pmr |
Normal distribution:
| 0 | ranking | 5 | ranking | 10 | ranking | 25 | ranking | 50 | ranking | 100 | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 95.61 | mhr | 87.77 | mhr | 74.22 | mhr | 29.60 | phr | 2.19 | vhr | 0 | vmr |
| 95.54 | vmr_phr | 85.46 | vmr_phr | 69.32 | vmr_phr | 25.16 | vhr | 2.08 | phr | 0 | vhr |
| 95.17 | mmr | 82.92 | vhr_pmr | 69.19 | phr | 21.63 | mhr | 0.40 | vmr | 0 | pmr |
| 94.25 | vhr_pmr | 82.78 | mmr | 62.77 | vhr_pmr | 13.98 | vmr | 0.21 | mhr | 0 | phr |
| 90.97 | pmr | 81.58 | phr | 62.19 | vhr | 13.74 | vmr_phr | 0.05 | vmr_phr | 0 | mmr |
| 90.17 | phr | 75.33 | pmr | 58.26 | mmr | 9.72 | vhr_pmr | 0.01 | vhr_pmr | 0 | mhr |
| 85.95 | vmr | 74.85 | vhr | 56.07 | vmr | 4.79 | pmr | 0.00 | pmr | 0 | vmr_phr |
| 84.93 | vhr | 72.53 | vmr | 52.17 | pmr | 4.08 | mmr | 0.00 | mmr | 0 | vhr_pmr |
| 200 | ranking | 300 | ranking | 400 | ranking | 500 | ranking | 1000 | ranking |
|---|---|---|---|---|---|---|---|---|---|
| 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr | 0 | vmr |
| 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr | 0 | vhr |
| 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr | 0 | pmr |
| 0 | phr | 0 | phr | 0 | phr | 0 | phr | 0 | phr |
| 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr | 0 | mmr |
| 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr | 0 | mhr |
| 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr | 0 | vmr_phr |
| 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr | 0 | vhr_pmr |
Summary statistics
Fit summary
Summary for fit of log returns to an F-S skew standardized Student-t
distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape
parameter.
xi is the estimated skewness parameter.
Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.005 | 0.007 | 0.005 | 0.008 | 0.005 | 0.007 | 0.007 | 0.006 |
| s | 0.027 | 0.034 | 0.019 | 0.030 | 0.023 | 0.031 | 0.028 | 0.027 |
| nu | 3.384 | 3.488 | 3.474 | 3.959 | 3.344 | 3.702 | 3.726 | 3.369 |
| xi | 0.699 | 0.708 | 0.770 | 0.737 | 0.716 | 0.714 | 0.715 | 0.709 |
| R^2 | 0.993 | 0.992 | 0.994 | 0.996 | 0.993 | 0.993 | 0.994 | 0.993 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.013 | 0.015 | 0.012 | 0.014 | 0.012 | 0.015 | 0.014 | 0.013 |
| s | 0.032 | 0.040 | 0.027 | 0.035 | 0.029 | 0.037 | 0.033 | 0.033 |
| nu | 3.446 | 3.510 | 2.629 | 4.002 | 3.035 | 3.780 | 3.760 | 3.260 |
| R^2 | 0.978 | 0.978 | 0.962 | 0.981 | 0.971 | 0.977 | 0.978 | 0.974 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| m | 0.006 | 0.007 | 0.005 | 0.008 | 0.006 | 0.008 | 0.007 | 0.006 |
| s | 0.024 | 0.031 | 0.017 | 0.028 | 0.021 | 0.029 | 0.026 | 0.024 |
| R^2 | 0.968 | 0.969 | 0.962 | 0.973 | 0.965 | 0.969 | 0.969 | 0.966 |
AIC and BIC
AIC
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| sstd | -671.412 | -603.343 | -768.368 | -622.474 | -718.914 | -617.207 | -647.901 | -672.555 |
| std | -628.069 | -561.055 | -728.782 | -585.156 | -674.537 | -574.620 | -605.443 | -628.239 |
| normal | -646.514 | -579.369 | -743.179 | -603.088 | -692.459 | -593.830 | -624.670 | -646.870 |
BIC
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| sstd | -659.588 | -591.520 | -756.545 | -610.651 | -707.091 | -605.384 | -636.077 | -660.732 |
| std | -616.246 | -549.232 | -716.959 | -573.333 | -662.714 | -562.796 | -593.620 | -616.416 |
| normal | -634.691 | -567.546 | -731.356 | -591.265 | -680.636 | -582.006 | -612.847 | -635.046 |
Kappa
Let \(\{X_{g,i}\}\) be Gaussian distributed with mean \(\mu\) and scale \(\sigma\).
Let \(\{X_{\nu,i}\}\) be \(t\)-distributed, scaled such that \(\mathbb{M}^{\nu}(1) = \mathbb{M}^{g}(1) = \sqrt{\frac{2}{\pi}} \sigma\).
Given \(n_g\), we want to determine and \(n_{\nu}^{*}\) such that
\[\text{Var}\left[\sum_i^{n_g} X_{g,i}\right] = \text{Var}\left[\sum_i^{n_{\nu}^{*}} X_{\nu,i}\right]\]
For iid. r.v \(\{X_i\}\):
\[S_n = X_1 + X_2 + \dots + X_n\] \[\mathbb{M}(n) = \mathbb{E}(\lvert S_n - \mathbb{E}(S_n)\rvert)\] Taleb’s convergence metric (\(\kappa\)):
The “rate” of convergence for \(n\) summands vs \(n_0\), i.e. the improved convergence achieved by \(n - n_0\) additional terms, is given by \(\kappa(n_0, n)\):
\[\kappa(n_0, n) = 2 - \dfrac{\log(n) - \log(n_0)}{\log\left(\frac{\mathbb{M}(n)}{\mathbb{M}(n_0)}\right)}\]
\(\kappa\)
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr |
|---|---|---|---|---|---|---|---|
| 0.16 | 0.16 | 0.16 | 0.13 | 0.15 | 0.14 | 0.13 | 0.17 |
\(n_{min}\)
What is the minimum value of \(n_{\nu}\), the number of observations from a given skewed \(t\)-distribution, we need to achieve the same degree of convergence as with \(n_g=30\) observations from a Gaussian distribution with the same mean and standard deviation?
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr |
|---|---|---|---|---|---|---|---|
| 58 | 55 | 55 | 48 | 58 | 51 | 50 | 57 |
Fit statistics ranking
Skewed \(t\)-distribution (sstd):
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.008 | phr | 0.019 | pmr | 0.996 | phr |
| 0.007 | mhr | 0.023 | mmr | 0.994 | vmr_phr |
| 0.007 | vmr_phr | 0.027 | vhr_pmr | 0.994 | pmr |
| 0.007 | vhr | 0.027 | vmr | 0.993 | mmr |
| 0.006 | vhr_pmr | 0.028 | vmr_phr | 0.993 | mhr |
| 0.005 | vmr | 0.030 | phr | 0.993 | vmr |
| 0.005 | pmr | 0.031 | mhr | 0.993 | vhr_pmr |
| 0.005 | mmr | 0.034 | vhr | 0.992 | vhr |
Standardized \(t\)-distribution (std):
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.015 | vhr | 0.027 | pmr | 0.981 | phr |
| 0.015 | mhr | 0.029 | mmr | 0.978 | vmr |
| 0.014 | phr | 0.032 | vmr | 0.978 | vhr |
| 0.014 | vmr_phr | 0.033 | vhr_pmr | 0.978 | vmr_phr |
| 0.013 | vhr_pmr | 0.033 | vmr_phr | 0.977 | mhr |
| 0.013 | vmr | 0.035 | phr | 0.974 | vhr_pmr |
| 0.012 | mmr | 0.037 | mhr | 0.971 | mmr |
| 0.012 | pmr | 0.040 | vhr | 0.962 | pmr |
Normal distribution:
| m | ranking | s | ranking | R^2 | ranking |
|---|---|---|---|---|---|
| 0.008 | phr | 0.017 | pmr | 0.973 | phr |
| 0.008 | mhr | 0.021 | mmr | 0.969 | vmr_phr |
| 0.007 | vhr | 0.024 | vhr_pmr | 0.969 | vhr |
| 0.007 | vmr_phr | 0.024 | vmr | 0.969 | mhr |
| 0.006 | vhr_pmr | 0.026 | vmr_phr | 0.968 | vmr |
| 0.006 | vmr | 0.028 | phr | 0.966 | vhr_pmr |
| 0.006 | mmr | 0.029 | mhr | 0.965 | mmr |
| 0.005 | pmr | 0.031 | vhr | 0.962 | pmr |
Monte Carlo simulations summary
Monte Carlo simulations of portfolio index values (currency
values).
Statistics are given for the final state of all paths.
Probability of down-and-out is calculated as the share of paths that
reach 0 at some point. All subsequent values for a path are set to 0, if
the path reaches at any point.
0 is defined as any value below a threshold.
dai_pct (for down-and-in) is the probability of losing
money. This is calculated as the share of paths finishing below index
100.
## Number of paths: 10000Skewed \(t\)-distribution (sstd):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 390.56 | 546.54 | 368.99 | 744.90 | 379.50 | 649.19 | 569.75 | 458.13 |
| mc_s | 162.67 | 310.60 | 106.60 | 358.90 | 98.41 | 230.77 | 195.66 | 154.95 |
| mc_min | 9.95 | 47.50 | 41.88 | 67.17 | 120.88 | 157.64 | 134.70 | 146.99 |
| mc_max | 1962.47 | 9418.03 | 964.46 | 3790.71 | 969.20 | 2350.77 | 2262.60 | 1819.14 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.32 | 0.32 | 0.03 | 0.03 | 0.00 | 0.00 | 0.00 | 0.00 |
Standardized \(t\)-distribution (std):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 2281.79 | 4769.87 | 1795.80 | 3664.55 | 2345.79 | 4253.11 | 2979.65 | 3285.27 |
| mc_s | 1214.65 | 3240.61 | 1239.38 | 2099.78 | 31176.36 | 3565.88 | 1263.15 | 1921.35 |
| mc_min | 221.36 | 374.70 | 41.43 | 193.68 | 557.15 | 783.35 | 689.69 | 499.82 |
| mc_max | 16109.79 | 45965.08 | 94908.01 | 28483.53 | 3118787.05 | 295755.16 | 30086.97 | 59511.22 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| mc_m | 441.92 | 627.78 | 365.42 | 789.64 | 403.75 | 711.18 | 614.07 | 498.41 |
| mc_s | 169.65 | 319.17 | 98.48 | 362.10 | 98.81 | 244.56 | 197.97 | 164.59 |
| mc_min | 90.51 | 80.75 | 123.03 | 120.90 | 165.90 | 188.76 | 136.86 | 160.53 |
| mc_max | 1791.46 | 3370.41 | 1109.65 | 3404.35 | 1102.37 | 2538.96 | 1961.78 | 1606.02 |
| dao_pct | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| dai_pct | 0.01 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Ranking
Skewed \(t\)-distribution (sstd):
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 744.90 | phr | 98.41 | mmr | 157.64 | mhr | 9418.03 | vhr | 0 | vmr | 0.00 | mmr |
| 649.19 | mhr | 106.60 | pmr | 146.99 | vhr_pmr | 3790.71 | phr | 0 | vhr | 0.00 | mhr |
| 569.75 | vmr_phr | 154.95 | vhr_pmr | 134.70 | vmr_phr | 2350.77 | mhr | 0 | pmr | 0.00 | vmr_phr |
| 546.54 | vhr | 162.67 | vmr | 120.88 | mmr | 2262.60 | vmr_phr | 0 | phr | 0.00 | vhr_pmr |
| 458.13 | vhr_pmr | 195.66 | vmr_phr | 67.17 | phr | 1962.47 | vmr | 0 | mmr | 0.03 | pmr |
| 390.56 | vmr | 230.77 | mhr | 47.50 | vhr | 1819.14 | vhr_pmr | 0 | mhr | 0.03 | phr |
| 379.50 | mmr | 310.60 | vhr | 41.88 | pmr | 969.20 | mmr | 0 | vmr_phr | 0.32 | vmr |
| 368.99 | pmr | 358.90 | phr | 9.95 | vmr | 964.46 | pmr | 0 | vhr_pmr | 0.32 | vhr |
Standardized \(t\)-distribution (std):
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 4769.87 | vhr | 1214.65 | vmr | 783.35 | mhr | 3118787.05 | mmr | 0 | vmr | 0.00 | vmr |
| 4253.11 | mhr | 1239.38 | pmr | 689.69 | vmr_phr | 295755.16 | mhr | 0 | vhr | 0.00 | vhr |
| 3664.55 | phr | 1263.15 | vmr_phr | 557.15 | mmr | 94908.01 | pmr | 0 | pmr | 0.00 | phr |
| 3285.27 | vhr_pmr | 1921.35 | vhr_pmr | 499.82 | vhr_pmr | 59511.22 | vhr_pmr | 0 | phr | 0.00 | mmr |
| 2979.65 | vmr_phr | 2099.78 | phr | 374.70 | vhr | 45965.08 | vhr | 0 | mmr | 0.00 | mhr |
| 2345.79 | mmr | 3240.61 | vhr | 221.36 | vmr | 30086.97 | vmr_phr | 0 | mhr | 0.00 | vmr_phr |
| 2281.79 | vmr | 3565.88 | mhr | 193.68 | phr | 28483.53 | phr | 0 | vmr_phr | 0.00 | vhr_pmr |
| 1795.80 | pmr | 31176.36 | mmr | 41.43 | pmr | 16109.79 | vmr | 0 | vhr_pmr | 0.01 | pmr |
Normal distribution:
| mc_m | ranking | mc_s | ranking | mc_min | ranking | mc_max | ranking | dao_pct | ranking | dai_pct | ranking |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 789.64 | phr | 98.48 | pmr | 188.76 | mhr | 3404.35 | phr | 0 | vmr | 0.00 | pmr |
| 711.18 | mhr | 98.81 | mmr | 165.90 | mmr | 3370.41 | vhr | 0 | vhr | 0.00 | phr |
| 627.78 | vhr | 164.59 | vhr_pmr | 160.53 | vhr_pmr | 2538.96 | mhr | 0 | pmr | 0.00 | mmr |
| 614.07 | vmr_phr | 169.65 | vmr | 136.86 | vmr_phr | 1961.78 | vmr_phr | 0 | phr | 0.00 | mhr |
| 498.41 | vhr_pmr | 197.97 | vmr_phr | 123.03 | pmr | 1791.46 | vmr | 0 | mmr | 0.00 | vmr_phr |
| 441.92 | vmr | 244.56 | mhr | 120.90 | phr | 1606.02 | vhr_pmr | 0 | mhr | 0.00 | vhr_pmr |
| 403.75 | mmr | 319.17 | vhr | 90.51 | vmr | 1109.65 | pmr | 0 | vmr_phr | 0.01 | vmr |
| 365.42 | pmr | 362.10 | phr | 80.75 | vhr | 1102.37 | mmr | 0 | vhr_pmr | 0.01 | vhr |
Compare Gaussian and skewed t-distribution fits
Gaussian fits
Gaussian QQ plots
Gaussian vs skewed t
Probability in percent that the smallest and largest (respectively) observed return for each fund was generated by a normal distribution:
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| P_norm(X_min) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| P_norm(X_max) | 0.546 | 0.580 | 1.599 | 0.796 | 1.161 | 0.746 | 0.793 | 0.903 |
| P_t(X_min) | 0.556 | 0.523 | 0.342 | 0.387 | 0.476 | 0.443 | 0.433 | 0.499 |
| P_t(X_max) | 0.448 | 0.469 | 1.135 | 0.614 | 0.739 | 0.518 | 0.543 | 0.613 |
Average number of years between min or max events (respectively):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| norm: avg yrs btw min | 1.157446e+77 | 3.872665e+82 | 3.449915e+57 | 4.076881e+55 | 2.913573e+67 | 1.048138e+67 | 2.446247e+63 | 4.894371e+71 |
| norm: avg yrs btw max | 1.830960e+02 | 1.724260e+02 | 6.252800e+01 | 1.256870e+02 | 8.614100e+01 | 1.341240e+02 | 1.261390e+02 | 1.107200e+02 |
| t: avg yrs btw min | 1.798360e+02 | 1.912300e+02 | 2.924120e+02 | 2.584920e+02 | 2.099400e+02 | 2.257430e+02 | 2.309140e+02 | 2.002190e+02 |
| t: avg yrs btw max | 2.233340e+02 | 2.130540e+02 | 8.811500e+01 | 1.628680e+02 | 1.352750e+02 | 1.928930e+02 | 1.843230e+02 | 1.630560e+02 |
Lilliefors test
p-values for Lilliefors test.
Testing \(H_0\), that log-returns are
Gaussian.
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| p value | 0 | 0 | 0 | 0 | 0 | 0.001 | 0.001 | 0 |
Wittgenstein’s Ruler
For different given probabilities that returns are Gaussian, what is the probability that the distribution is Gaussian rather than skewed t-distributed, given the smallest/largest observed log-returns?
Conditional probabilities for smallest observed log-returns:
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \min(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \leq x_{\text{min}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0.000 | 0.00 | 0.000 | 0.00 | 0.000 | 0.001 | 0.001 | 0.000 |
| Prior prob | 1.000 | 1.00 | 1.000 | 1.00 | 1.000 | 0.999 | 0.999 | 1.000 |
| P[Gauss | Event] | 0.838 | 0.82 | 0.768 | 0.69 | 0.815 | 0.486 | 0.418 | 0.965 |
Use \(1 - \text{p-value}\) from
Lilliefors test as prior probability that the distribution is
Gaussian.
\(x_{\text{obs}} = \max(x)\) and \(P[\text{Event}\ |\ \text{Gaussian}] =
P_{\text{Gauss}}[X \geq x_{\text{max}}]\):
| vmr | vhr | pmr | phr | mmr | mhr | vmr_phr | vhr_pmr | |
|---|---|---|---|---|---|---|---|---|
| Lillie p-val | 0 | 0 | 0 | 0 | 0 | 0.001 | 0.001 | 0 |
| Prior prob | 1 | 1 | 1 | 1 | 1 | 0.999 | 0.999 | 1 |
| P[Gauss | Event] | 1 | 1 | 1 | 1 | 1 | 1.000 | 1.000 | 1 |
Velliv medium risk (vmr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.4145605 0.3807834Objective function plots
Velliv high risk (vhr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.7391222 0.4858909Objective function plots
PFA medium risk (pmr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.3304634 0.2764028Objective function plots
PFA high risk (phr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
phr has the sstd fit with the highest sstd fit with thevalue of nu. Compare with other distributions:
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 2.0162301 0.4463226Objective function plots
Mix medium risk (mmr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
mmr has the sstd fit with the lowest value of nu. Compare with other distributions:
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.3516393 0.2503782Objective function plots
Mix high risk (mhr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.8775189 0.3400818Objective function plots
Mix vmr+phr (vm_ph), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.7507161 0.3263777Objective function plots
Mix vhr+pmr (mh_pm), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.540135 0.320090Objective function plots
Velliv medium risk (vmr), June 2012 - April 2024
QQ Plot
Skewed \(t\)-distribution (sstd):
Data vs fit
Let’s plot the fit and the observed returns together.
Estimated distribution
Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:
Monte Carlo
Sorted portfolio index values for last period of all runs
Convergence
Max vs sum
Max vs sum plots for the first four moments:
MC
IS
Skewed \(t\)-distribution with a normal proposal distribution.
Parameters
## [1] 1.4145605 0.3807834Objective function plots